The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2012) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.
Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number.
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